... vertex x
of degree 6. Thus V is a disjoint union of the closed neighborhood N[x]ofxand
a set A of cardinality 6. Consider the 6 × 12 matrix M whose rows are indexed
by the elements of A and the columns ... the subgraph of G induced by A.
Table 1 contains the value of γ(n, δ) for n ≤ 16, 0 ≤ δ ≤ n − 1 if the value is
known, otherwise upper and lower bounds fo...
... approximations of the roots for
specific values of d are found easily, and so are the inverses of the roots of
smallest modulus giving us the entries of first line of table 2.
the electronic journal of combinatorics ... denotes the identity
matrix and A is the state transfer matrix of the automaton. Consequently,
the root of smallest modulus of the deno...
... definition of δ
k
(n). The theorem
therefore follows.
In the next section we show that if k = 3 then the above theorem holds for n ≥ 6.
Note that the results in this section are stated for fixed ... asterisks. The values given for these cases are the
best known upper bounds.
the electronic journal of combinatorics 7 (2000), #R58 14
Lemma 4 If there esists an (n, k, 1)...
... spoke. As the set of spokes form an edge-cut of the graph W (n, 2k +3), such
a cycle must in fact use an even number of spokes. If the number of spokes used is four
or more then the number of vertices ... preceding it in the clockwise orientation and q the distance along the rim
from n
p,κ
. We place the following restriction on the spokes of the wheel, indicat...
... 1) for 1 k ⌊
n−1
2
⌋
(v) µ(n, 4, 3)
n(n − 1)
3
for n 4
The following theorem provides upper bounds for µ(n, d, w) even if w > d.
Theorem 9. For all n 3,
the electronic journal of ... [15], many of the upper bounds obtained by linear pro-
gramming coincide with the bound µ(n, d)
n!
(d−1)!
of theorem 1. For 14 n 16,
computations of the LP b...
... (14) is the best in all known upper
bounds for G
2
, and bound (18) is the best in all known upper bounds for G
3
. Finally,
bound (22) is the best upper bound for G
1
and bound (1 8) is the second-best ... new and sharp upper bounds for
λ(G), which are better than all of the above mentioned upper bounds in some sense, and
determine the extremal g r aph...
... T
κ
.
2.22
The proof of Corollary 2.6 is complete.
Remark 2.7. The result of Theorem 2.5 holds for an arbitrary time scale. Therefore, using
Theorem 2.5, we can obtain many results for some peculiar ... follows, R denotes the set of real numbers, R
0, ∞, Z denotes the set of integers,
N
0
{0, 1, 2, } denotes the set of nonnegative integers, CM, S denotes th...
... when the number of relays is relatively large,
the gain of ergodic capacity is co nstant, approximately.
Take the symmetric channels as example, at SNR = 10
dB, the gain of the upper bound of ergodic ... exploited.
For the investigation of TWOR-NC-Dir systems, the
canonical argument line [18] is that we must first obtain
the statistical description of the ins...
... best; next is the upper WS bound followed by
the upper C bound. For λ
2
(Q), the SS bounds are the same
as the C bounds. For λ
3
(Q), the upper SS bound is the best;
next is the upper WS bound ... bounds of the eigenvalues of Q are functions of
ρ in the interval [0, 1]. Therefore, we are able to find optimal
bounds of the eigenvalues of Q by c...
... condition I
. Therefore, by Theorem 2.4,from3.7, we easily obtain the
estimate 3.3 of solutions of 3.1. The proof of Theorem 3.2 is complete.
Using Theorem 2.5, we easily obtain the following ... T
0
. 2.25
Therefore, the desired inequality 2.6 follows from 2.10, 2.22,and2.25. This completes
the proof.
Theorem 2.5. Suppose that all assumptions of Theorem...